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Defects in photonic crystals are local regions in which the translational symmetry is broken. The same definition can be applied to photonic quasicrystals except in this case the symmetry is the 2π/n rotational symmetry, where n is the rotational fold number. In this context, if no such defects are present, the structure is called “defect-free”. Even though photonic quasicrystal patterns can be defect-free, localized modes can still exist in such structures. These modes resemble those of a central potential that suggests that localization in photonic quasicrystals are actually “extended” modes of the rotational symmetry. A possible connection is suggested between these localized modes and short-range dependence of the photonic band gap (PBG). Such a connection implies a tight-binding description of PBG formation of photonic quasicrystals ‒ making them more similar to electronic semiconductors than regular photonic crystals. Physical coupling to these defect-free localized modes is demonstrated.
©2007 Optical Society of America
1. Introduction
Photonic crystals (PhCs) have been an active area of research for nearly two decades [1,2]. The reason for this interest stems from PhCs possessing a bandstructure for photons in more than one direction. This implies that they may potentially be the elemental components of all-optical circuitry [3]. This bandstructure comes about from the spatial periodicity of the dielectric scatterers that make up PhCs. Omnidirectional band gaps, known as photonic band gaps (PBGs), may also be present within these bandstructures. Spatial periodicity, however, is not necessarily a requirement for generating photonic bandstructures. Motivated by the discovery of electronic quasicrystals [4], Chan et al. showed that 2D Penrose-tiled photonic quasicrystals (PhQs) also have photonic bandstructures, and in some cases, their PBGs were larger and more isotropic [5]. In PhCs, defects are introduced by locally breaking the spatial periodicity. This results in optical modes becoming localized to these areas. The same concepts apply to PhQs, however, the symmetry that is locally broken is the 2π/n rotational symmetry, where n is the rotational fold number. Optical modes will also localize to these defects in PhQs and in fact have been shown to provide richer and more wavelength selective defect states [6] when compared to PhC defects. An important discovery by Wang et al. [7] showed that defect states occurred in defect-free dodecagonal 2D Penrose-tiled PhQs, although these defect states did not occur for octagonal and decagonal PhQs. Borrowing an analogous rationalization from electronic quasicrystals, they explained these results by a competition between the self-similarity of quasi-periodic patterns and local nonperiodicity. Recently [8], this explanation was disputed with an alternate theory that states that localization occurs because of nearest-neighbor resonances and that scatterer distribution symmetry constitutes the most favorable localization conditions. There seems to be some ambiguity in the literature concerning the formation of these localized modes. In addition to this, PBG formation in PhQs is also unclear. As pointed out by Della Villa et al. [9], there seems to be contributions from both long and short-range effects for Penrose-tiled PhQs. Gauthier et al. [10] postulated a possible link between localized modes and the effect of short-and long-range influences on PBG formation. By proposing a new type of PhQ and demonstrating that the PBG of these PhQs had a short-range dependence in addition to the usual long-range dependence, they showed that by considerably reducing the sample size of this PhQ, the PBGs remained with one important addition: defect states were now present within the PBGs. Closer examination of these states showed that they were in fact localized modes and they possessed symmetries similar to the local structure of the PhQ. This has been shown already for a 10-fold Penrose type PhQ [11]. It seems likely that there is a relationship between these short-range mechanisms that form PBGs in PhQs and localized states in defect-free PhQs.
In this paper, we will investigate these localized modes further using a finite-element method (FEM) and finite-difference time-domain approach. It will be shown that a group of these localized modes is very similar to the modes of a central potential and that they can be described as possessing an orbital angular momentum number equal to the rotational fold number of the PhQ. It seems very reasonable that the short-range dependence of the PBG in PhQs is related to a selection rule governing these localized modes. The long-range order of these localization sites gives rise to the long-range dependence of the PBG. We physically demonstrate the reality of these localized modes via optical coupling from a waveguide near these localization sites.
2. Theory and Simulation
The PhQs proposed by Gauthier et al. were produced from a dual-beam holographic lithography process [12]. It involves generating a group of overlapped plane families that have a pre-determined angle between their respective normals. This angle sets the rotational symmetry of the PhQ pattern. One of these plane families can represent a single exposure in some photosensitive media. If the threshold exposure level of this media were larger than any single exposure, the overlapped areas of the different plane families would cumulatively add until this threshold level is reached. This threshold level would depend upon the number of plane families required. Having reached the threshold, the material would be fully exposed in those regions and after a development step, one would then have the resulting PhQ pattern. The density, size, and shape of these regions (i.e. the dielectric scatterers) depend upon this threshold. This process provides an efficient way of building PhQs with geometrically correlated dielectric scatterers, which provides an extra degree of freedom when studying PhQ bandstructures [13] and localized modes. Fig. 1 shows this process for a 12-fold PhQ. In this case, six plane families are required with a 30° angle between the respective normals. This method can also generate regular PhCs. Square and hexagonal lattices correspond to 4- and 6-fold patterns. For details on using this method for physical fabrication see Gauthier et al. [14].
Fig. 1. Six plane families generating a 12-fold PhQ. The intensity threshold correlates the density, size and shape of the scatterers. (a) With an intensity threshold set to 66% of the total intensity, a sparse distribution of almost circular scatterers appears, while (b) if the intensity threshold is set to 50%, a denser distribution of various shaped scatterers appears.
Theoretical tools for analyzing the bandstructure of PhQs are not as developed as for those used in PhCs. Methods such as planewave expansions are ill defined and require large primitive cells which are computational exhaustive. A popular technique for analyzing PhQs has been the multiple expansion method [15]. This method calculates the impulse response (i.e. Green’s function) of the differential equation governing the dynamics of the field quantity. The form of the relation is,
where Ĥ can be either the electric or magnetic field operator, G(r, ω) is the Fourier transformed Green’s function and δ(r - r 0) is the Dirac delta function. If the dielectric scatterer shapes are circular cylinders, then predetermined impulse responses, such as Hankel functions, form the basis of the solution. The bandstructure can then be interpreted from the local density of states (LDOS), which is simply the trace of the imaginary part of the Green’s function, G(r,ω),
This technique becomes difficult, if not impossible, to apply, if the scatterers are not well-defined geometrical shapes [16]. The shape of the dielectric scatterers is very important to the photonic bandstructure. Base states that combine to make the available modes are selected by the physical structure of the dielectric ‒ similar to how the atomic form factor affects bandstructure in solid-state physics. We call the physical structure of the dielectric the dielectric form factor (DFF). Because of the difficulty that analytical and semi-analytical methods have in dealing with non-standard dielectric shapes, two numerical techniques are used: the finite-difference time-domain (FDTD) method, which is well equipped in finding the transmission spectra of complicated PhQ structures [10], and the finite-element method (FEM), which is appropriate for analyzing and isolating the eigenmodes of complicated shapes, will be used in this paper.
Fig. 2 shows the results of two FDTD transmission spectra of the same PhQ pattern (Fig. 1b) but with two difference widths: the top being 10 μm wide and the bottom being 3 μm wide. Propagation is in the direction of the width. The 10 μm wide PhQ has a rather large stop region in the larger wavelengths and a few smaller ones in the lower wavelength region. Reducing the width gave rise to defect states in these stop regions. It should be noted clearly that these defect states were not caused by any physical defects within the structure but occurred simply by changing the width of the structure.
Fig. 2. Comparison of transmission data for 10 μm (upper plot) and 3 μm (lower plot). As the width changed, defect states appeared in the smaller width PhQ.
It appears that local regions within a PhQ have their own bandstructures and only when these regions come together, one obtains the full picture of the overall PhQ bandstructure. This implies then that some localized modes are dependent only on their immediate region. In order to study these localized modes in a more detailed manner, an eigenvalue analysis was required. Applying FEM to this 12-fold PhQ showed a variety of localized mode profiles that resemble the modes of a central potential. They resemble spherical harmonic functions that depend upon the order of rotational symmetry of the PhQ and the scatterer shapes. For example, in this case, the 12-fold PhQ would have an orbital angular momentum number l = 12, implying that it has 2l + 1 = 25 localized modes that can be realized by the DFF. Each mode is represented by the states (l,m) where m = 0, ±1, ±2,⋯. Figure 3 shows the results of the FEM analysis.
Fig. 3. Localized optical modes in a 12-fold PhQ. The number of negative lobes (represented in red and a negative sign) or the number of positive lobes (represented in blue and a positive sign) equals the m number. The wavelength at which these modes occur are shown underneath each respective mode.
The number of blue or red lobes within the region of the central rod equals the m number. Each of the images in Fig. 3 comes with another image that has the same m number, but slightly rotated. These would correspond to the negative m numbers. These localized modes resemble angular momentum modes of a central potential. These results imply that there seems to be a selection rule that combines the PhQ’s crystal angular momentum (i.e. the rotational symmetry of the PhQ lattice) and the dielectric form factor that chooses the localized mode. Wang et al. did not find localized modes for octagonal and decagonal (8-fold and 10-fold, respectively) Penrose PhQs. The 8-fold and 10-fold non-Penrose PhQs used in this paper also did not have localized modes when the scatterers were circular. However, non-circular scatterer shapes allowed some of these localized modes to appear. Results are shown in Fig. 4 and Fig. 5.
Fig. 4. Localized modes in 8-fold PhQs when the dielectric scatterer shapes are different from circular.
Fig. 5. Localized modes in 10-fold PhQs when the dielectric scatterer shapes are different from circular.
One way of describing this selection process is as follows: the dielectric distribution aids in selecting available modes from the completeness of spherical harmonics in space and manifests them by making them available for real optical modes. This process describes the physical manifestation of expanding the optical modes in terms of spherical harmonics and a radial function that depends upon the dielectric:
where 〈r|ψ〉 is the scalar transverse electric field in Dirac notation, 〈l,m|ψ〉 is the component projecting the field value on angular momentum modes, 〈l,0|Rz(ϕ)|l,m〉 are the spherical harmonic functions in Dirac notation and 〈r|l,0〉 is the radial dependence term that is directly related to the DFF.
In the next section, this selection process using the DFF is physically demonstrated when a silicon photonic wire waveguide is passed through a 12-fold PhQ pattern and the light in this guide couples to this one of these localized modes.
3. Measurement
The above simulations indicate that these defect-free localized modes are independent of long-range interactions. One can physically demonstrate this by altering the dielectric near the sites where these localized modes appear. By placing a dielectric waveguide near one of these sites, as shown in fig. 6, the localized mode is still present but the resonant wavelength is slightly different.
Using microfabrication techniques, a converted scanning electron microscope wrote the 12-fold PhQ pattern in an electron-beam sensitive resist. The resist acted as a mask for an electron-cyclotron resonance (ECR) etch. After cleaving, the structure was prepared for measurement. Fig. 7 shows two transmission spectra. The upper spectrum (a) was a broad sweep of a straight waveguide with no PhQ and the straight waveguide near the PhQ. The lower spectrum was a higher resolution scan of the large dip region of the upper spectrum. The overall drop in intensity in the waveguide near the PhQ was due to an inefficient interface between the waveguide and the PhQ region. However, the prominent dip near 1480 nm can only be due to coupling from the waveguide to the localized mode. Fig. 8(b–d) show three overhead infrared images at three different wavelengths. There is no taper region between the input waveguide and the waveguide in the pattern. This results in a large amount of scattered light and poor coupling efficiency as seen to the left hand side of the structure trace. By enhancing the images with false color, however, it is easy to see the localized mode. The trace of the physical structure has been added to aid in viewing. The images in Fig. 8b and Fig. 8c show no large intensity profile from the center of the structure. However, Fig. 8d does show a relatively large intensity profile at the center of the structure at the dip wavelength. Fig. 9 is a FEM simulation of (a) the electric field and (b) the time-averaged power at the dip wavelength. These results show a strong agreement between simulation and measurement.
Fig. 7. (a) Transmission spectra of PhQ compared with straight waveguide. Losses are high to design roughness, but notice the large drop around the resonant wavelength for the localized mode. (b) Higher resolution spectra scan for localized mode.
Fig. 8. (a) The physical picture of the 12-fold PhQ and the waveguide setup. (b) and (c) Intensity pictures of the scattering at off-dip wavelength values. The arrows point to weak scattering in the area of the center of the pattern. (d) Intensity picture of the localized mode shown exactly where the dip occurs. The dashed box represents the outline of quasicrystal pattern.
4. Conclusion
In conclusion, the short-range dependence of localized modes in defect-free PhQs has been physically confirmed via experimental analysis and comparison to simulations. The localization method in PhQs however is fundamentally different from disorder structures. These localized modes are very similar to the modes of a central potential. With this being the case, it is possible to propose that the localization process that occurs in PhQs may in fact stem from a crystal angular momentum selection rule that arises from the rotational symmetry of the PhQs. These localized modes can be considered to be the “extended” modes of the rotational symmetries making them appear as localizations. PhQs have many regions that have a large rotational symmetry. The short-range dependence of PhQ PBG band gaps may in fact stem from these localized modes. The overall contribution of all the sites in the PhQ will give rise to long-range dependences of the PhQ PBG. This description is similar to the tight-binding description in solid-state theory. The localized modes can be seen as optical atoms, and these optical atoms interact to make the larger bandstructure. One can then propose that the PhQ is actually closer in representing an optical semiconductor than a regular PhC.
Acknowledgments
The authors would like to thank NSERC, NRC and especially Edith Post for her help on measurement phase of the experimental work.
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